#5220 carg, polarform and rectform of a product do not reduce the argument to (-%pi, %pi]

[David Scherfgen]

Bug report written by Claude.

absarg in src/rpart.lisp computes the argument of a product as the sum of the arguments of its factors and never reduces the sum into (-%pi, %pi]. A power does get reduced, by a ceiling term, a product does not. So carg(-x) is atan2(0,x)+%pi, which is 2*%pi for a negative x, and every function built on absarg that takes a root of the argument returns a wrong value for such x: rectform(sqrt(-x)) is -2^(3/2) at x = -8, where sqrt(8) is 2^(3/2).

(%i1) display2d : false$
(%i2) carg(-x);
(%o2) atan2(0,x)+%pi
(%i3) subst(x = -8, %);
(%o3) 2*%pi
(%i4) carg(-x*y);
(%o4) atan2(0,y)+atan2(0,x)+%pi
(%i5) subst([x = -1, y = -1], %);
(%o5) 3*%pi
(%i6) rectform(sqrt(-x));
(%o6) %i*sin((atan2(0,x)+%pi)/2)*sqrt(abs(x))+cos((atan2(0,x)+%pi)/2)*sqrt(abs(x))
(%i7) subst(x = -8, %);
(%o7) -2^(3/2)
(%i8) sqrt(-x), x = -8;
(%o8) 2^(3/2)
(%i9) domain : complex$
(%i10) rectform((-x)^(1/3));
(%o10) %i*sin((atan2(0,x)+%pi)/3)*abs(x)^(1/3)+cos((atan2(0,x)+%pi)/3)*abs(x)^(1/3)
(%i11) subst(x = -8, %);
(%o11) sqrt(3)*%i-1
(%i12) float(rectform((-x)^(1/3))), x = -8;
(%o12) 2.0
(%i13) carg(x^3);
(%o13) 3*atan2(0,x)-2*%pi*ceiling((3*atan2(0,x)-%pi)/(2*%pi))

The last output shows the reduction that a power gets and a product lacks.

Expected: carg(-x) equal to atan2(0, -x), that is 0 for x < 0 and %pi for x > 0 (for instance %pi - atan2(0, x)), carg(-x*y) equal to %pi at x = y = -1, rectform(sqrt(-x)) equal to 2^(3/2) at x = -8, and rectform((-x)^(1/3)) equal to 2 there with domain : complex.

Where: absarg in src/rpart.lisp, the mtimes clause, which returns (2pistrip (addn argl t)); 2pistrip strips only exact multiples of 2*%pi and leaves a sum such as atan2(0,x)+%pi alone.